# Everyday Life As A Double Dual

> Shariputra, form does not differ from emptiness, emptiness does
> not differ from form. What is form is emptiness, what is emptiness
> is form. The same is true of sensations, perceptions, conception
> and knowledge.
>
> Heart Sutra

> Before one studies Zen, mountains are mountains and waters are
> waters; after a first glimpse into the truth of Zen, mountains
> are no longer mountains and waters are no longer waters; after
> enlightenment, mountains are once again mountains and waters once
> again waters.
>
> Dougen

Though I'm not a Buddhist myself.



In math, there are two different things called point and function.
A point is a point. A function is what maps a point to something.
If it maps to a scalar, it is called a functional. If you don't
know what a scalar is, just think of it as a number now.

But after you reach some level, you start to think that a point
also maps a functional to a scalar. Like this:

a(f) = f(a)
a(x^2 - x + 1) = a^2 - a + 1

Note that I abused the notation. I should use a** on the left-hand
side instead of a. To make it explicit that it is the double dual
of a.


I'm not going to explain what duality exactly means. If I explain
it rigorously, this post will not end. So to explain it extremely
roughly, the dual of a point is a functional.

a** is the double dual here. It is the dual of the dual of a point.
It takes a functional and maps it to a scalar. It is a functional
for functionals.

As you can guess from its name, the original space and its double
dual is NATURALLY isomorphic. That means the two are structurally
totally the same. So the abuse of the notation is quite justified.
a = a**.

Point and function are not two different things. You can regard it
as a point or a function as you want. But since you can't easily
calculate points unlike functions, dealing with functions is usually
easier.

There is also a pairing e(a,f) = a(f) = f(a). Where form and matter
meet. Integration is also a kind of pairings. The dual of (1,0),
(0,1) are dx, dy.


An advanced extension of this line of thinking is the Yoneda lemma.

Nat(Hom(A,-), F) ~= F(A)

You could guess the meaning by seeing the notations. It literally
means form and emptiness are the same. The two are naturally
isomorphic. I think Cayley's theorem shows this the most intuitively.



Everyday life is naturally isomorphic to its double dual. The forms
for forms. Then why should I strictly distinguish matter from forms?
It's relative.

I don't accept Aristotle's hylomorphism. Forms independently exist.
Though they should meet matter to create. Instead I have started
to think that all thinkable things exist. This might be the dual
of Buddhist thinking that everything is emptiness.

The world is mathematically designed and follows the fate determined
by the planets. I don't distinguish the real mathematical world
from the unreal shadows. There are simply beautiful and ugly parts
of the world. Mountains are mountains. Waters are waters.

The world might be also naturally isomorphic to its double dual.
I'm not fully convinced though. So that the world precedes and is
preceded by god at the same time. God has to create the world in
order to become god. God is transcendental. But god's energies
shine via the beauty of the world.

This is also similar to Animus in Anima. Masculine in Feminine in
Masculine. A matryoshka. Maybe the feminine comes first if you
think of the virgin birth. Or the masculine comes first if you
think of nous. However god is androgyne. That guarantees the
transcendence of god.

3=1+2 reduces to 1.